Journal of Lie Theory
Volume 14 (2004) 523–535
c 2004 Heldermann Verlag
On Dimension Formulas for gl(m|n) Representations
E. M. Moens and J. Van der Jeugt
Communicated by H. Schlosser
Abstract.
We investigate new formulas for the dimension and superdimension of covariant representations Vλ of the Lie superalgebra gl(m|n) . The notion
of t -dimension is introduced, where the parameter t keeps track of the Z-grading
of Vλ . Thus when t = 1, the t -dimension reduces to the ordinary dimension,
and when t = −1 it reduces to the superdimension. An interesting formula
for the t -dimension is derived from a recently obtained new formula for the supersymmetric Schur polynomial sλ (x/y), which yields the character of Vλ . It
expresses the t -dimension as a simple determinant. For a special choice of λ ,
the new t -dimension formula gives rise to a Hankel determinant identity.
1.
Introduction
Let g be the Lie superalgebra gl(m|n). The general linear Lie superalgebra is
one of the standard families of classical Lie superalgebras. Lie superalgebras are
characterized by a Z2 -grading g = g0̄ ⊕ g1̄ . For the general theory on classical Lie
superalgebras and their representations, we refer to [5, 6, 16].
Let h ⊂ g be the Cartan subalgebra of g, and g = g−1 ⊕ g0 ⊕ g+1 be the Zgrading that is consistent with the Z2 -grading of g. Note that g0 = g0̄ = gl(m) ⊕
gl(n). The dual space h∗ of h has a natural basis {1 , . . . , m , δ1 , . . . , δn }, and the
roots of g can be expressed in terms of this basis. We shall work here with the socalled distinguished choice [5] for a triangular decomposition of g. In that case, the
positive even roots are given by {i − j |1 ≤ i < j ≤ m} ∪ {δi − δj |1 ≤ i < j ≤ n},
and the positive odd roots by {i − δj |1 ≤ i ≤ m, 1 ≤ j ≤ n}.
Representation theory of Lie superalgebras, and in particular of gl(m|n) or
its simple counterpart sl(m|n), is not a straightforward copy of the corresponding
theory for simple Lie algebras. It is mainly due to the existence of atypical
representations [6] that problems occur [22, 23, 24], in particular to compute
the character. Only recently a solution has been proposed to some of these
problems [17, 3] for gl(m|n). In this paper, however, we shall be dealing only with
the so-called covariant representations of gl(m|n), for which an explicit character
formula is known.
Let V be a finite-dimensional irreducible representation of g. Such modules
are h-diagonalizable with weight decomposition V = ⊕µ V (µ), and the character is
c Heldermann Verlag
ISSN 0949–5932 / $2.50 524
Moens and Van der Jeugt
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defined to be ch V = µ dim V (µ) eµ , where eµ (µ ∈ h∗ ) is the formal exponential.
Let Λ be the highest weight of V . We shall consider the specialization of ch V
determined by
F (ei ) = 1 (i = 1, . . . , m)
(1)
F (eδj ) = t (j = 1, . . . , n).
This specialization is consistent with the Z-grading of g, and the corresponding
Z-grading of V . The specialization of the character of V under F is referred to
as the t-dimension of V and denoted by dim t (V ):
X
dim t (V ) = F (ch V ) =
dim V (µ)F (eµ ).
(2)
µ
Often, the t-dimension would be defined [7, §10] as F (e−Λ ch V ), with Λ the
highest weight of V ; but here (2) is more convenient. The t-dimension of V
stands for the polynomial
X
F (eΛ )
dim V−j tj ,
(3)
j∈Z+
where V = V0 ⊕ V−1 ⊕ V−2 ⊕ · · · is the Z-grading of V . Note that for the Z2 grading V = V0̄ ⊕ V1̄ we have V0̄ = V0 ⊕ V−2 ⊕ · · · and V1̄ = V−1 ⊕ V−3 ⊕ · · · .
Therefore, the dimension of V is found by putting t = 1 in the expression for the
t-dimension, whereas the superdimension of V is found by putting t = −1. So
the t-dimension can also be seen as an extension of the notion of dimension and
superdimension.
This paper is dealing with the computation of the t-dimension of a particular class of finite-dimensional irreducible representations of gl(m|n), namely the
covariant representations. These were introduced by Berele and Regev [2], and
Sergeev [18]. They showed that the tensor product of N copies of the natural
(m + n)-dimensional representation of gl(m/n) is completely reducible, and that
the irreducible components Vλ can be labeled by a partition λ of N such that λ
is inside the (m, n)-hook, i.e. such that λm+1 ≤ n. Berele and Regev not only
introduced these representations, they also gave a character formula for them. The
character of Vλ is known as a supersymmetric Schur function [2, 9, 20]. It is a
polynomial in variables xi (i = 1, . . . , m) and yj (j = 1, . . . , n) with xi = ei and
yj = eδj , and denoted by
ch Vλ = sλ (x/y).
(4)
There exist a number of expressions for sλ (x/y). One is a combinatorial
expression, by means of supertableaux [2, 9]. Another expression is a formula
due to Sergeev and Pragacz [14, 21, 15]. These two formulas, however, are less
convenient to determine the t-dimension. In order to compute the t-dimension,
there are two useful formulas. The first is the classical formula relating the supersymmetric Schur function sλ (x/y) to the determinant of elementary or complete
supersymmetric polynomials. These formulas go back to [4, 1], see also [9]. The
second is a new determinantal formula for supersymmetric Schur polynomials [12].
For the first formula, consider the complete supersymmetric functions defined by
r
X
hr (x/y) =
hr−k (x)ek (y),
(5)
k=0
Moens and Van der Jeugt
525
where hr−k and ek are the complete and elementary symmetric functions [11]
respectively. Then the supersymmetric Schur polynomial is given by
sλ (x/y) = det
hλi −i+j (x/y) ,
(6)
1≤i,j≤`(λ)
where ` ≡ `(λ) is the length of the partition λ = (λ1 , λ2 , . . . , λ` ). The polynomials
sλ (x/y) are identically zero when λm+1 > n.
Since xi = ei and yj = eδj , the specialization (1) corresponds to putting
each xi = 1 and yj = t in sλ (x/y). For the elementary and complete symmetric
functions, such specializations are well-known:
m+r−1
m+r−1
hr (x1 , . . . , xm )
=
=
,
(7)
r
m−1
xi =1
m
er (x1 , . . . , xm )
=
.
(8)
r
xi =1
Thus it follows from (5) and (6) that
Proposition 1.1.
The t-dimension of Vλ is given by the determinant
!
λi −i+j X m + λi − i + j − k − 1n
dim t Vλ = det
tk .
1≤i,j≤`(λ)
λi − i + j − k
k
k=0
(9)
Although this formula is simple to derive, it should be observed that in
general the matrix elements in the right hand side of (9) do not have a “closed
form”
[13]:
they remain polynomials in t. Even for t = 1, the expression
Pr expression
m+r−k−1 n
cannot be simplified in general. Only for t = −1 we have
k=1
r−k
k
r X
m+r−k−1 n
k=0
r−k
k
k
(−1) =
m−n−1+r
.
r
This is related to the fact that
r X
m+r−1
−r, −n
m+r−k−1 n k
; −t ,
t =
2 F1
r
−m − r + 1
r−k
k
k=0
(10)
in terms of the 2 F1 hypergeometric function [13, 19], and the terminating 2 F1
series – with general parameters – is summable only with argument 1.
This implies that for t = −1, i.e. the superdimension formula sdim Vλ , the
expression (9) can be simplified:
m − n − 1 + λi − i + j
sdim Vλ =
det
(11)
1≤i,j≤`(λ)
λi − i + j
Q
Y
i<j (λi − i − λj + j)
Q
=
(m − n + 1 − i)λi .
(12)
i (λi − i + l(λ))
i
Herein, (a)n = a(a + 1) · · · (a + n − 1) is the Pochhammer symbol [19], and the
determinant in (11) can be written in closed form using [10, (3.11)]. So in general
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Moens and Van der Jeugt
the superdimension has a closed form expression (12), whereas the dimension has
not.
Observe that (12) yields: if m ≤ n then sdim Vλ = 0 when λ1 + m > n and
sdim Vλ 6= 0 when λ1 + m ≤ n; if m > n then sdim Vλ = 0 when λ01 + n > m and
sdim Vλ 6= 0 when λ01 + n ≤ m (where λ0 is the conjugate of λ).
In the following section we shall consider the new determinantal formula for
supersymmetric Schur polynomials [12], and use it to compute the t-dimension.
This time, the expression for dim t (Vλ ) is quite different from (9): it reduces again
to a determinant, but now the matrix elements are closed forms in t instead of
hypergeometric series in t. We shall then simplify this expression, and discuss
some applications.
2.
A formula for the t-dimension
The starting point of our new t-dimension formula is the recently introduced determinantal formula for the supersymmetric Schur function sλ (x/y) [12], deduced
using a character formula of Kac and Wakimoto [8]. Let x = x(m) = (x1 , . . . , xm )
and y = y (n) = (y1 , . . . , yn ); let λ be a partition with λm+1 ≤ n (i.e. λ is inside
the (m, n)-hook), and let k be the (m, n)-index of λ:
k = min{j|λj + m + 1 − j ≤ n};
(13)
see [12] for its meaning: in particular, m − k + 1 is the atypicality of the representation Vλ . As usual, λ0 denotes the conjugate of λ. The new formula reads:


λj +m−n−j
1
x
i
xi +yj 1≤i≤m
1≤i≤m

1≤j≤k−1 
sλ (x/y) = ±D−1 det  λ0 +n−m−i 1≤j≤n
 (14)
yj i
0
1≤i≤n−m+k−1
1≤j≤n
with
Q
D=
Q
− xj ) i<j (yi − yj )
Q
.
i,j (xi + yj )
i<j (xi
Observe that the sign in (14) is (−1)mn−m+k−1 ; since its role is not essential here,
we shall usually just write ±.
In order to deduce a t-dimension formula from (14) we will need some simple
properties of symmetric polynomials and a careful analysis of the determinant
in (14) using row and column operations.
We have already mentioned the complete and elementary symmetric functions. Another class that we need here are the monomial symmetric functions
mλ (x) [11]. The number of terms in mλ (x) is easy to count, so that we have the
following counterpart of (8):
m(0r0 1r1 ...krk ) (x1 , . . . , xm )
xi =1
k
X
m!
=
where
ri = m.
r0 !r1 ! . . . rk !
i=0
(15)
The following lemma gives some simple decomposition properties of symmetric functions:
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Moens and Van der Jeugt
Lemma 2.1.
Let x = x0 + x00 be a decomposition of x = (x1 , . . . , xm ) in two
disjoint subsets. Then
hr (x) =
r
X
X
hk (x0 )hr−k (x00 ) and mλ (x) =
k=0
mµ (x0 )mν (x00 ).
µ∪ν=λ
Proof.
The proof for the hr (x) polynomials follows immediately from the generating function for these polynomials [11, (I.2.5)]. For the mλ , we use induction
on |x00 |. First, let x00 = (xm ). By the definition of mλ (x), with λ = (λ1 , λ2 , . . .),
if follows that
X
X
mλ (x) = mλ (x0 ) +
mµ (x0 )mλi (xm ) =
mµ (x0 )mν (xm ).
λi ∪µ=λ
µ∪ν=λ
Now assume that the property holds for |x00 | ≤ q . Let x̄0 = x0 \ {xi } and
x̄00 = x00 ∪ {xi } for a certain xi ∈ x0 . Then, using the induction hypothesis:
XX
X
0
0
00
mµ (x̄ )mη (xi ) mκ (x00 )
mτ (x )mκ (x ) =
mλ (x) =
τ ∪κ=λ
τ ∪κ=λ
=
X
µ∪η=τ
X
X
00
0
mµ (x̄0 )mν (x̄00 ).
mη (xi )mκ (x ) =
mµ (x̄ )
µ∪ν=λ
η∪κ=ν
µ∪ν=λ
Next, we shall use a number of times the same sequence of elementary row
or column operations in matrices. So it is convenient to fix these in an algorithm:
Algorithm 1 Given a matrix with at least m rows, with Ri denoting row i. The
algorithm consists of the following row operations:
Step 1:
Step 2:
..
.
Step m − 1 :
Ri − R1
,
xi − x1
Ri − R2
,
Ri −→
xi − x2
Ri −→
Rm −→
for 1 < i ≤ m;
for 2 < i ≤ m;
Rm − Rm−1
.
xm − xm−1
So the total number of row operations is m(m − 1)/2.
Algorithm 2 Given a matrix with at least n columns, with Cj denoting column
j . This algorithm consists of the following n(n − 1)/2 column operations:
Step 1:
Step 2:
..
.
Step n − 1 :
Cj
yj
Cj
Cj −→
yj
Cj −→
Cn −→
− C1
,
− y1
− C2
,
− y2
Cn − Cn−1
.
yn − yn−1
for 1 < j ≤ n;
for 2 < j ≤ n;
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Moens and Van der Jeugt
Lemma 2.2.
sider matrices
Let (r1 , r2 , . . .) be a sequence of (non-negative) integers, and con
A=
hrj (xi )
,
B=
1≤i≤p
1≤j≤q
hri (yj )
.
1≤i≤p
1≤j≤q
Then Algorithm 1 transforms A into A? , and Algorithm 2 transforms B into B ? ,
with
?
?
A = hrj −i+1 (x1 , . . . , xi )
,
B = hri −j+1 (y1 , . . . , yj )
.
1≤i≤p
1≤j≤q
1≤i≤p
1≤j≤q
Proof.
It is sufficient to give the proof for A only (so we assume p ≥ m).
Denote by A(s) the matrix obtained after step s of the algorithm. We shall prove
(s)
that the (i, j)-element of A(s) is given by Ai,j = hrj −s (x1 , . . . , xs , xi ), by induction
on s. Clearly, in the first step the elements hrj (xi ) are replaced by
r
r
xi j − x1j
r −1
r −2
r −2
r −1
= xi j + xi j x1 + . . . + xi x1j + x1j = hrj −1 (x1 , xi ).
xi − x1
(s)
Now we can assume that after step s we have Ai,j = hrj −s (x1 , . . . , xs , xi ) for all
i > s. Step s + 1 consist of the operations Ri −→ (Ri − Rs+1 )/(xi − xs+1 ) for all
(s+1)
i > s + 1. Thus the element Ai,j becomes, using Lemma 2.1 a number of times:
hrj −s (x1 , . . . , xs , xi ) − hrj −s (x1 , . . . , xs , xs+1 )
=
xi − xs+1
rj −s−1
r −s−l
rj −s−l
X
xi j
− xs+1
=
hl (x1 , . . . , xs )
xi − xs+1
l=0
rj −s−1
=
X
r −s−l−1
hl (x1 , . . . , xs )(xi j
r −s−l−2
+ xi j
xs+1 + . . .
l=0
r −s−l−2
j
+xi xs+1
r −s−l−1
j
+ xs+1
)
rj −s−1
=
X
hl (x1 , . . . , xs )hrj −s−l−1 (xs+1 , xi ) = hrj −s−1 (x1 , . . . , xs+1 , xi ).
l=0
Since the algorithm applies in total i − 1 row transformations on row i, it follows
that A?i,j = hrj −i+1 (x1 , . . . , xi ).
Lemma 2.3.
Algorithm 1 transforms R =
?
R =
(−1)i−1
Qi
l=1 (xl + yj )
1
xi + yj
.
1≤i≤m
1≤j≤n
into
1≤i≤m
1≤j≤n
529
Moens and Van der Jeugt
Proof.
Denote by R(s) the matrix obtained after step s of the algorithm. We
shall prove that the (i, j)-element of R(s) is given by
(−1)s
.
l=1 (xl + yj )(xi + yj )
(s)
Ri,j = Qs
−R1
In the first step, the operations are Ri −→ Rxii −x
for i > 1, so
1
1
1
−1
1
(1)
Ri,j =
−
=
.
xi + yj
x1 + yj xi − x1
(x1 + yj )(xi + yj )
Next we use induction on s. One finds:
(−1)s
(−1)s
1
(s+1)
Qs
Ri,j
=
− Qs
l=1 (xl + yj )(xi + yj )
l=1 (xl + yj )(xs+1 + yj ) xi − xs+1
s
(−1)
(−1)
= Qs
·
.
l=1 (xl + yj ) (xs+1 + yj )(xi + yj )
Since the algorithm applies in total i − 1 row transformations on row i, the result
follows.
The following is a technical lemma on partitions, using the reverse lexicographic ordering [11, §I.1] for partitions of the same integer. So when we write
λ ≤ µ, this means that λ and µ are partitions of the same integer (i.e. |λ| = |µ|)
with either λ = µ or else the first non-vanishing difference λi − µi negative.
Lemma 2.4.
Assume that α, β, ν, µ are partitions with `(α) = s + 1, `(β) =
s + 2 and `(ν) = 2. Then, for i, s, t ∈ N:
α ≤ (i, 1s ),
µ ∪ (t) = α,
ν ≤ (t, 1)
β = µ ∪ ν ≤ (i, 1s+1 ).
⇔
Proof.
Assume that α ≤ (i, 1s ), µ ∪ (t) = α and ν ≤ (t, 1), then |β| =
|µ| + |ν| = (i + s − t) + (t + 1) = |(i, 1s+1 )|. Furthermore β1 = max(µ1 , ν1 ) ≤
max(µ1 , t) = α1 ≤ i, so β ≤ (i, 1s+1 ).
Conversely, assume that β = µ ∪ ν ≤ (i, 1s+1 ), then ν is of the form ν = (βk , βl )
(βl > 0), so ν ≤ (βk + βl − 1, 1). Put t = βk + βl − 1 and α = µ ∪ (t). Then
|α| = |µ| + |(t)| = (i + s + 1 − βk − βl ) + (βk + βl − 1) = i + s. Since `(µ) = s
we have that |µ| ≥ s, and |(t)| ≤ i. So α1 = max(µ1 , t) ≤ max(µ1 , i) ≤ i, thus
α ≤ (i, 1s ).
This technical lemma is needed in the following:
Lemma 2.5.
Let Yj =
R=
1
1+yj
and consider the matrix
(−1)i+1
(1 + yj )i
Algorithm 2 transforms R into
?
R = (−1)i+j
=
1≤i≤m
1≤j≤n
X
α≤(i,1j−1 )
i+1
(−1)
Yji
.
1≤i≤m
1≤j≤n
mα (Y1 , . . . , Yj )
.
1≤i≤p
1≤j≤n
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Moens and Van der Jeugt
Proof.
Observe that 1/(yi − yj ) = Yi Yj /(Yj − Yi ). Denote, as usual, by R(s)
the matrix obtained after step s of the algorithm. We shall prove that
X
(s)
Ri,j = (−1)i+s+1
mα (Y1 , . . . , Ys , Yj ),
for all j > s.
α≤(i,1s )
Step 1 consist of the column operations Cj −→
Cj − C1
(1)
Y1 Yj , so Ri,j is given by
Y1 − Yj
Y1 Yj
Y1 − Yj
i−1 2
i+2
i
= (−1) (Yj Y1 + Yj Y1 + . . . + Yj2 Y1i−1 + Yj Y1i )
X
= (−1)i+2
mα (Y1 , Yj ).
i+1
(−1)
Yji
i+1
− (−1)
Y1i
α≤(i,1)
Next we use induction on s. This yields, using Lemma 2.1:
(s+1)
Ri,j
X i+s+1
= (−1)
α≤(i,1s )
X
= (−1)i+s+1
Yj Ys+1
mα (Y1 , . . . , Ys , Yj ) − mα (Y1 , . . . , Ys , Ys+1 )
Ys+1 − Yj
X
t
mµ (Y1 , . . . , Ys )(Yjt − Ys+1
)
α≤(i,1s ) µ∪(t)=α
X
= (−1)i+s+2
X
mµ (Y1 , . . . , Ys )
α≤(i,1s ) µ∪(t)=α
X
Yj Ys+1
Ys+1 − Yj
mν (Ys+1 , Yj ).
ν≤(t,1)
Next, we use Lemma 2.4 and finally Lemma 2.1 again:
X X
(s+1)
i+s+2
Ri,j
= (−1)
mµ (Y1 , . . . , Ys )mν (Ys+1 , Yj )
β≤(i,1s+1 )
X
= (−1)i+s+2
µ∪ν=β
mβ (Y1 , . . . , Ys+1 , Yj ).
β≤(i,1s+1 )
Since the algorithm applies in total j − 1 column transformations on column j ,
the result follows.
The next lemma is about the specialization of such matrix elements. By
y = 1 we mean the substitution (y1 = 1, . . . , yj = 1).
Lemma 2.6.
X
Ri,j =
α≤(i,1j−1 )
Proof.
Ri,j
mα (y1 , . . . , yj )
=
y=1
i+j−2
.
j−1
It is easy to verify (e.g. using (15)) that R1,j = 1 and Ri,1 = 1. Now,
X
=
mα (y1 , . . . , yj )
y=1
α≤(i,1s+1 )
=
X
mµ (y1 , . . . , yj−1 ) yj +
µ≤(i,1j−2 )
= Ri,j−1 + Ri−1,j .
Hence the result follows.
X
ν≤(i−1,1j−1 )
mν (y1 , . . . , yj ) yj y=1
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Moens and Van der Jeugt
Now we have all ingredients to determine the specialization of (14).
The t-dimension of Vλ is given by dim t (Vλ ) = ±(1 + t)mn R(λ)
Theorem 2.7.
with


R(λ) = det  (−1)i+j
i+j−2
(1+t)i+j−1 j−1
λ0i +n−m−i−j+1
t
λj +m−n−j
i−1
1≤i≤m
1≤j≤n
λ0 +n−m−i
1≤i≤m
1≤j≤k−1

.
0
i
j−1

1≤i≤n−m+k−1
1≤j≤n
(16)
Proof.
Consider the determinant in (14) and apply Algorithm 1 on the corresponding matrix. From Lemmas 2.2 and 2.3 it follows that the first m rows of this
matrix become
(−1)i−1
Qi
) 1≤i≤m
hλj +m−n−i−j+1 (x1 , . . . , xi ) 1≤i≤m
(x +y )
l=1
l
j
1≤j≤k−1
1≤j≤n
while the determinant has been multiplied by a factor
make the substitution xi = 1; then (14) becomes

(−1)i−1
Q
m
i
(1+yj )
1≤i≤m

j (1 + yj )
±Q
det  λ0 +n−m−i 1≤j≤n
yj i
i<j (yi − yj )
1≤i≤n−m+k−1
Q
i>j (xi
− xj ). Now we can
λj +m−n−j
i−1

1≤i≤m
1≤j≤k−1
0

.
1≤j≤n
Next apply Algorithm 2 on the first n columns of this matrix. Using Lemmas 2.2
and 2.5, this becomes



X
λj +m−n−j
(−1)i+j

mα (Y1 , . . . , Yj )
Y
1≤i≤m
i−1

m
1≤j≤k−1 
(1 + yj ) det
.
α≤(i,1j−1 )
1≤i≤m


1≤j≤n
j
hλ0i +n−m−i−j+1 (y1 , . . . , yj ) 1≤i≤n−m+k−1
0
1≤j≤n
Finally, substituting yj = t, using Lemma 2.6, and the fact that we are dealing
with homogeneous symmetric polynomials, leads to the result.
Compared to (9), (16) has the advantage that each matrix element is a
simple binomial coefficient multiplied by a power of t or (1 + t), and no longer a
finite series of type 2 F1 (−t). So in general (16) is easier to compute. Furthermore,
its simple form is more appropriate to deduce certain properties of the t-dimension
for particular Vλ , as we shall demonstrate in the following section.
3.
Further simplifications, examples and applications
Let λ be a partition in the (m, n)-hook, and λ0 its conjugate. Recall the definition
of the (m, n)-index k of λ in (13), and let us also define the related integer r :
k = min{i|λi + m + 1 − i ≤ n},
r = n − m + k − λk − 1.
(1 ≤ k ≤ m + 1);
532
Moens and Van der Jeugt
For the combinatorial meaning of r , see [12]. Since λ is in the (m, n)-hook, λ0 is
in the (n, m)-hook, and we can define its (n, m)-index k 0 and the corresponding
number r0 :
k 0 = min{i|λ0i + n + 1 − i ≤ m},
(1 ≤ k 0 ≤ n + 1);
r0 = m − n + k 0 − λ0k0 − 1.
Applying the determinant formula (14) for sλ (x/y) and for sλ0 (y/x) yields
the same, with determinants of transposed matrices. Comparing the orders of the
matrices implies that n + k − 1 = m + k 0 − 1, so we have
n + k = m + k0,
r = k 0 − λk − 1,
r0 = k − λ0k0 − 1.
(17)
Furthermore, from [12, Lemma 3.2] we know that λ0λk +l = k − 1 for all 1 ≤ l ≤ r .
So the binomials on the last r rows of the matrix in (16) take the values
0
λi + n − m − i
r−l
=
for 1 ≤ l ≤ r, and i = λk + l.
j−1
j−1
By the triangularity of the matrix with such binomial coefficients as entries, the
determinant in (16) can thus be reduced according to the last r rows.
Completely analogous, the remaining determinant can be reduced according
to the last r0 columns. What remains is the determinant of a matrix of order
n + k − 1 − r − r0 , and we have
Corollary 3.1.
The t-dimension of Vλ is given by dim t (Vλ ) = ±(1+t)mn R0 (λ)
with

 i+j+r+r 0
0

R0 (λ) = det  (−1)
i+j+r+r −2
j−1
(1+t)i+j+r+r0 −1
λ0i +n−m−i−j−r+1
t
1≤i≤m−r 0
1≤j≤n−r
λ0 +n−m−i
i
j+r−1
1≤i≤λk
1≤j≤n−r
λj +m−n−j
i+r 0 −1
1≤i≤m−r 0
1≤j≤λ0 0
k
0


(18)
An interesting
application follows from this formula for the special case of
(m−a)
λ = (n − a)
, where a = 0, 1, . . . , min(m, n). For such a rectangular λ, we
have
k = m − a + 1, k 0 = n − a + 1, r = n − a, r0 = m − a, λk = 0, λ0k0 = 0,
and so the determinant in (18) reduces:
dim t (Vλ )
0
(−1)i+j+r+r
i + j + r + r0 − 2
= ±(1 + t)
det
1≤i,j≤a
(1 + t)i+j+r+r0 −1
j−1
i+j
(−1)
i + j + m + n − 2a − 2
mn−a(r+r0 −1)
= ±(1 + t)
det
,
1≤i,j≤a
(1 + t)i+j
j−1
mn
The resulting determinant can be further simplified: in the corresponding matrix,
multiply row i by (−1)i+1 (1 + t)i+1 for all 1 ≤ i ≤ a, and then multiply column
j by (−1)1+j (1 + t)j−1 for all 1 ≤ j ≤ a. This yields:
i + j + m + n − 2a − 2
(m−a)(n−a)
dim t (Vλ ) = (1 + t)
det
.
1≤i,j≤a
j−1
533
Moens and Van der Jeugt
Now the matrix elements have no longer a power of (1 + t), but only a binomial
coefficient. The remaining determinant can easily be computed. Taking out
common factors in rows and columns, it becomes
a
Y
i=1
(i + m + n − 2a − 1)!
det (m + n − 2a + i)j−1 .
(i + n − a − 1)!(i + m − a − 1)! 1≤i,j≤a
The last determinant is of the form
det (xi )j−1 = det
1≤i,j≤a
1≤i,j≤a
xj−1
=
i
Y
(xj − xi ),
1≤i<j≤a
see [10, (2.2)].
So we finally obtain, for λ = (n − a)(m−a) , that
a−1
Y
(m + n − 2a + i)! i!
(n − a + i)!(m − a + i)!
i=0
a−1
Y m+n−2a+i
n−a+i
= (1 + t)(m−a)(n−a)
m−a+i
dim t (Vλ ) = (1 + t)(m−a)(n−a)
i=0
i
Comparing this with (9), we
form expression for determi obtain a closed
(m−a)
nants of the type (9) where λ = (n − a)
. Replacing m by m + 1, m − a
by s, and reversing the order of the rows of the corresponding matrix, this yields,
using the 2 F1 notation:
m − n − i − j, −n
det
; −t
2 F1
0≤i,j≤s
−n − i − j
m−s
Y 2s+n−m+i
s+1
= (−1)s(s+1)/2 (1 + t)(s+1)(s+n−m)
(s ≤ m).
s+i
n+i+j
m
i=1
s+1
The change of order of the rows implies we are dealing with a Hankel determinant,
and for such determinants the row and column indices are usually starting from
0. This determinant identity can be written in a number of alternative ways. E.g.
applying a transformation on the 2 F1 , and denoting t/(t + 1) by z , one can write
s(s+1)/2
det (Ai+j ) = (−1)
0≤i,j≤s
(s+1)(m−s)
(1 − z)
m−s
Y
i=1
2s+n−m+i
s+1
s+i
s+1
,
(19)
where
Ak =
n+k
m
2 F1
−m, −n
;z .
−n − k
(20)
Since this is a polynomial identity in n, the condition that n must be an integer
can be dropped. Replacing n by u and z by −v , one can write this in the following
form:
534
Moens and Van der Jeugt
Corollary 3.2.
variables, and
Let m and s be positive integers with s ≤ m, u and v arbitrary
m X
u+k−l u
Ak =
l=0
m−l
l
vl .
Then the Hankel determinant is given by
s(s+1)/2
det (Ai+j ) = (−1)
0≤i,j≤s
(s+1)(m−s)
(1 + v)
m−s
Y
i=1
2s+u−m+i
s+1
s+i
s+1
.
It seems to be difficult to find an independent proof of this corollary, even
with the methods of [10, §2.6]. Here, it is a simple consequence of the two different
t-dimension formulas for a particular Vλ .
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Balantekin, A. B., and I. Bars, Dimension and character formulas for Lie
supergroups, Journal of Mathematical Physics 22 (1981), 1149–1162.
Berele, A., and A. Regev, Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras, Advances in Mathematics 64 (1987), 118–175.
Brundan, J., Kazhdan-Lusztig polynomials and character formulae for the
Lie superalgebra gl(m|n), Journal of the American Mathematical Society
16 (2003), 185–231.
Dondi, P. H., and P. D. Jarvis, Diagram and superfield techniques in the
classical superalgebras, Journal of Physics A 14 (1981), 547–563.
Kac, V. G., Lie superalgebras, Advances in Mathematics 26 (1977), 8–96.
—, Representations of classical Lie superalgebras, Lecture Notes in Mathematics 676, 597–626, Springer, Berlin, 1978.
—, “Infinite dimensional Lie algebras,” Cambridge University Press, Cambridge, 1990.
Kac, V. G., and M. Wakimoto, Integrable highest weight modules over
affine superalgebras and number theory, Progress in Mathematics 123
(1994), 415–456.
King, R. C., Supersymmetric functions and the Lie supergroup U (m/n),
Ars Combinatoria 16 A (1983), 269–287.
Krattenthaler, C., Advanced determinant calculus, Séminaire Lotharingien
Combinatoire 42 (1999), Article B42q, 67 pp.
Macdonald, I.G., “Symmetric functions and Hall polynomials, ” Oxford
University Press, Oxford, 2nd edition, 1995.
Moens, E. M., and J. Van der Jeugt, A determinantal formula for supersymmetric Schur polynomials, Journal of Algebraic Combinatorics 17
(2003), 283–307.
Petkovšek, M., H. S. Wilf and D. Zeilberger, “A=B”, A. K. Peters, Wellesley, Massachusetts, 1996.
Moens and Van der Jeugt
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
535
Pragacz, P., Algebro–geometric applications of Schur S- and Q-polynomials, Lecture Notes in Mathematics 1478, pp. 130–191, Springer, Berlin,
1991.
Pragacz, P., and A. Thorup, On a Jacobi-Trudi identity for supersymmetric polynomials, Advances in Mathematics 95 (1992), 8–17.
Scheunert, M., “The theory of Lie superalgebras”, Springer, Berlin, 1979.
Serganova, V., Kazhdan-Lusztig polynomials and character formula for the
Lie superalgebra gl(m|n), Selecta Mathematica 2 (1996), 607–651.
Sergeev, A. N., Tensor algebra of the identity representation as a module
over the Lie superalgebras Gl(n, m) and Q(n) (Russian), Matematicheskiı̆
Sbornik 123 (1984), 422–430.
Slater, L. J., “Generalized Hypergeometric Functions, ” Cambridge University Press, Cambridge, 1966.
Stembridge, J., A characterization of supersymmetric polynomials, Journal
of Algebra 95 (1985), 439–444.
Van der Jeugt, J., and V. Fack, The Pragacz identity and a new algorithm
for Littlewood-Richardson coefficients, Computers and Mathematics with
Applications 21 (1991), 39–47.
Van der Jeugt, J., J. W. B. Hughes, R. C. King and J. Thierry-Mieg, Character formulas for irreducible modules of the Lie superalgebra sl(m/n),
Journal of Mathematical Physics 31 (1990), 2278–2304.
—, A character formula for singly atypical modules of the Lie superalgebra
sl(m/n), Communications in Algebra 18 (1990), 3453–3480.
Van der Jeugt, J., and R. B. Zhang, Characters and composition factor
multiplicities for the Lie superalgebra gl(m/n), Letters in Mathematical
Physics 47 (1999), 49–61.
E. M. Moens and J. Van der Jeugt
Department of Applied Mathematics
and Computer Science
Ghent University
Krijgslaan 281-S9
B-9000 Gent
Belgium
ElsM.Moens@UGent.be
Joris.VanderJeugt@UGent.be
Received June 10, 2003
and in final form June 10, 2003